Principles of composite material mechanics, third edition (1)

Such knowledge is essential for advanced composite materials to be further applied to lightweight structures successfully.” —Nobuo Takeda, University of Tokyo, Japan Principles of Compo

“Professor Ron Gibson provides a comprehensive textbook to cover the

basic concept of anisotropy and inhomogeneity of composites as well as

state-of-art issues such as dynamic behavior, fracture and testing The many

worked-out examples and homework problems are most useful for readers

to understand the basic concepts to be used in practical applications of

composites Such knowledge is essential for advanced composite materials

to be further applied to lightweight structures successfully.”

—Nobuo Takeda, University of Tokyo, Japan

Principles of Composite Material Mechanics, Third Edition presents

a unique blend of classical and contemporary mechanics of composites

technologies While continuing to cover classical methods, this edition

also includes frequent references to current state-of-the-art composites

technology and research findings

New to the Third Edition

• Many new worked-out example problems, homework problems,

figures, and references

• An appendix on matrix concepts and operations

• Coverage of particle composites, nanocomposites, nanoenhancement

of conventional fiber composites, and hybrid multiscale composites

• Expanded coverage of finite element modeling and test methods

Easily accessible to students, this popular bestseller incorporates the most

worked-out example problems and exercises of any available textbook on

mechanics of composite materials It offers a rich, comprehensive, and

up-to-date foundation for students to begin their work in composite

materials science and engineering

MATERIAL MECHANICS

Gibson

PRINCIPLES OF COMPOSITE

MATERIAL

MECHANICS

THIRD EDITION

MECHANICAL ENGINEERING

A Series of Textbooks and Reference Books

Founding Editor

L L Faulkner

Columbus Division, Battelle Memorial Institute

and Department of Mechanical Engineering

The Ohio State University Columbus, Ohio

1 Spring Designer’s Handbook, Harold Carlson

2 Computer-Aided Graphics and Design, Daniel L Ryan

3 Lubrication Fundamentals, J George Wills

4 Solar Engineering for Domestic Buildings, William A Himmelman

5 Applied Engineering Mechanics: Statics and Dynamics, G Boothroydand C Poli

6 Centrifugal Pump Clinic, Igor J Karassik

7 Computer-Aided Kinetics for Machine Design, Daniel L Ryan

8 Plastics Products Design Handbook, Part A: Materials and

Components; Part B: Processes and Design for Processes, edited byEdward Miller

9 Turbomachinery: Basic Theory and Applications, Earl Logan, Jr

10 Vibrations of Shells and Plates, Werner Soedel

11 Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni

12 Practical Stress Analysis in Engineering Design, Alexander Blake

13 An Introduction to the Design and Behavior of Bolted Joints,

John H Bickford

14 Optimal Engineering Design: Principles and Applications,

James N Siddall

15 Spring Manufacturing Handbook, Harold Carlson

16 Industrial Noise Control: Fundamentals and Applications, edited byLewis H Bell

17 Gears and Their Vibration: A Basic Approach to Understanding GearNoise, J Derek Smith

18 Chains for Power Transmission and Material Handling: Design and Applications Handbook, American Chain Association

19 Corrosion and Corrosion Protection Handbook, edited by

Philip A Schweitzer

20 Gear Drive Systems: Design and Application, Peter Lynwander

21 Controlling In-Plant Airborne Contaminants: Systems Design

and Calculations, John D Constance

22 CAD/CAM Systems Planning and Implementation, Charles S Knox

23 Probabilistic Engineering Design: Principles and Applications, James N Siddall

24 Traction Drives: Selection and Application, Frederick W Heilich III and Eugene E Shube

25 Finite Element Methods: An Introduction, Ronald L Huston

and Chris E Passerello

26 Mechanical Fastening of Plastics: An Engineering Handbook,

Brayton Lincoln, Kenneth J Gomes, and James F Braden

27 Lubrication in Practice: Second Edition, edited by W S Robertson

28 Principles of Automated Drafting, Daniel L Ryan

29 Practical Seal Design, edited by Leonard J Martini

30 Engineering Documentation for CAD/CAM Applications,

and Mark E Coticchia

34 Steam Plant Calculations Manual, V Ganapathy

35 Design Assurance for Engineers and Managers, John A Burgess

36 Heat Transfer Fluids and Systems for Process and Energy

Applications, Jasbir Singh

37 Potential Flows: Computer Graphic Solutions, Robert H Kirchhoff

38 Computer-Aided Graphics and Design: Second Edition, Daniel L Ryan

39 Electronically Controlled Proportional Valves: Selection and

Application, Michael J Tonyan, edited by Tobi Goldoftas

40 Pressure Gauge Handbook, AMETEK, U.S Gauge Division, edited by Philip W Harland

41 Fabric Filtration for Combustion Sources: Fundamentals and BasicTechnology, R P Donovan

42 Design of Mechanical Joints, Alexander Blake

43 CAD/CAM Dictionary, Edward J Preston, George W Crawford, and Mark E Coticchia

44 Machinery Adhesives for Locking, Retaining, and Sealing,

Girard S Haviland

45 Couplings and Joints: Design, Selection, and Application,

Jon R Mancuso

46 Shaft Alignment Handbook, John Piotrowski

47 BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer, V Ganapathy

48 Solving Mechanical Design Problems with Computer Graphics, Jerome C Lange

49 Plastics Gearing: Selection and Application, Clifford E Adams

50 Clutches and Brakes: Design and Selection, William C Orthwein

51 Transducers in Mechanical and Electronic Design, Harry L Trietley

52 Metallurgical Applications of Shock-Wave and High-Strain-RatePhenomena, edited by Lawrence E Murr, Karl P Staudhammer, and Marc A Meyers

53 Magnesium Products Design, Robert S Busk

54 How to Integrate CAD/CAM Systems: Management and Technology, William D Engelke

55 Cam Design and Manufacture: Second Edition; with cam design software for the IBM PC and compatibles, disk included,

Preben W Jensen

56 Solid-State AC Motor Controls: Selection and Application,

Sylvester Campbell

57 Fundamentals of Robotics, David D Ardayfio

58 Belt Selection and Application for Engineers, edited by

Wallace D Erickson

59 Developing Three-Dimensional CAD Software with the IBM PC,

C Stan Wei

60 Organizing Data for CIM Applications, Charles S Knox, with

contributions by Thomas C Boos, Ross S Culverhouse,

and Paul F Muchnicki

61 Computer-Aided Simulation in Railway Dynamics, by Rao V Dukkipati and Joseph R Amyot

62 Fiber-Reinforced Composites: Materials, Manufacturing, and Design,

P K Mallick

63 Photoelectric Sensors and Controls: Selection and Application, Scott M Juds

64 Finite Element Analysis with Personal Computers,

Edward R Champion, Jr and J Michael Ensminger

65 Ultrasonics: Fundamentals, Technology, Applications: Second Edition,Revised and Expanded, Dale Ensminger

66 Applied Finite Element Modeling: Practical Problem Solving forEngineers, Jeffrey M Steele

67 Measurement and Instrumentation in Engineering: Principles and BasicLaboratory Experiments, Francis S Tse and Ivan E Morse

68 Centrifugal Pump Clinic: Second Edition, Revised and Expanded, Igor J Karassik

69 Practical Stress Analysis in Engineering Design: Second Edition,Revised and Expanded, Alexander Blake

70 An Introduction to the Design and Behavior of Bolted Joints:

Second Edition, Revised and Expanded, John H Bickford

71 High Vacuum Technology: A Practical Guide, Marsbed H Hablanian

72 Pressure Sensors: Selection and Application, Duane Tandeske

73 Zinc Handbook: Properties, Processing, and Use in Design,

Frank Porter

74 Thermal Fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski

75 Classical and Modern Mechanisms for Engineers and Inventors, Preben W Jensen

76 Handbook of Electronic Package Design, edited by Michael Pecht

77 Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by Marc A Meyers, Lawrence E Murr, and Karl P Staudhammer

78 Industrial Refrigeration: Principles, Design and Applications,

P C Koelet

79 Applied Combustion, Eugene L Keating

80 Engine Oils and Automotive Lubrication, edited by Wilfried J Bartz

81 Mechanism Analysis: Simplified and Graphical Techniques,

Second Edition, Revised and Expanded, Lyndon O Barton

82 Fundamental Fluid Mechanics for the Practicing Engineer,

86 Vibrations of Shells and Plates: Second Edition, Revised and

Expanded, Werner Soedel

87 Steam Plant Calculations Manual: Second Edition, Revised andExpanded, V Ganapathy

88 Industrial Noise Control: Fundamentals and Applications,

Second Edition, Revised and Expanded, Lewis H Bell

and Douglas H Bell

89 Finite Elements: Their Design and Performance, Richard H MacNeal

90 Mechanical Properties of Polymers and Composites: Second Edition,Revised and Expanded, Lawrence E Nielsen and Robert F Landel

91 Mechanical Wear Prediction and Prevention, Raymond G Bayer

92 Mechanical Power Transmission Components, edited by

David W South and Jon R Mancuso

93 Handbook of Turbomachinery, edited by Earl Logan, Jr

94 Engineering Documentation Control Practices and Procedures, Ray E Monahan

95 Refractory Linings Thermomechanical Design and Applications, Charles A Schacht

96 Geometric Dimensioning and Tolerancing: Applications and

Techniques for Use in Design, Manufacturing, and Inspection, James D Meadows

97 An Introduction to the Design and Behavior of Bolted Joints:

Third Edition, Revised and Expanded, John H Bickford

98 Shaft Alignment Handbook: Second Edition, Revised and Expanded,John Piotrowski

99 Computer-Aided Design of Polymer-Matrix Composite Structures,edited by Suong Van Hoa

100 Friction Science and Technology, Peter J Blau

101 Introduction to Plastics and Composites: Mechanical Properties and Engineering Applications, Edward Miller

102 Practical Fracture Mechanics in Design, Alexander Blake

103 Pump Characteristics and Applications, Michael W Volk

104 Optical Principles and Technology for Engineers, James E Stewart

105 Optimizing the Shape of Mechanical Elements and Structures,

A A Seireg and Jorge Rodriguez

106 Kinematics and Dynamics of Machinery, Vladimír Stejskal

and Michael Valásek

107 Shaft Seals for Dynamic Applications, Les Horve

108 Reliability-Based Mechanical Design, edited by Thomas A Cruse

109 Mechanical Fastening, Joining, and Assembly, James A Speck

110 Turbomachinery Fluid Dynamics and Heat Transfer, edited by

Chunill Hah

111 High-Vacuum Technology: A Practical Guide, Second Edition,

Revised and Expanded, Marsbed H Hablanian

112 Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D Meadows

113 Handbook of Materials Selection for Engineering Applications,

116 Applied Computational Fluid Dynamics, edited by Vijay K Garg

117 Fluid Sealing Technology, Heinz K Muller and Bernard S Nau

118 Friction and Lubrication in Mechanical Design, A A Seireg

119 Influence Functions and Matrices, Yuri A Melnikov

120 Mechanical Analysis of Electronic Packaging Systems,

Stephen A McKeown

121 Couplings and Joints: Design, Selection, and Application,

Second Edition, Revised and Expanded, Jon R Mancuso

122 Thermodynamics: Processes and Applications, Earl Logan, Jr

123 Gear Noise and Vibration, J Derek Smith

124 Practical Fluid Mechanics for Engineering Applications,

John J Bloomer

125 Handbook of Hydraulic Fluid Technology, edited by George E Totten

126 Heat Exchanger Design Handbook, T Kuppan

127 Designing for Product Sound Quality, Richard H Lyon

128 Probability Applications in Mechanical Design, Franklin E Fisher and Joy R Fisher

129 Nickel Alloys, edited by Ulrich Heubner

130 Rotating Machinery Vibration: Problem Analysis and Troubleshooting, Maurice L Adams, Jr

131 Formulas for Dynamic Analysis, Ronald L Huston and C Q Liu

132 Handbook of Machinery Dynamics, Lynn L Faulkner and Earl Logan, Jr

133 Rapid Prototyping Technology: Selection and Application,

136 Practical Guide to Industrial Boiler Systems, Ralph L Vandagriff

137 Lubrication Fundamentals: Second Edition, Revised and Expanded,

D M Pirro and A A Wessol

138.Mechanical Life Cycle Handbook: Good Environmental Design

and Manufacturing, edited by Mahendra S Hundal

139 Micromachining of Engineering Materials, edited by Joseph McGeough

140 Control Strategies for Dynamic Systems: Design and Implementation, John H Lumkes, Jr

141 Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot

142 Nondestructive Evaluation: Theory, Techniques, and Applications,edited by Peter J Shull

143 Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and Control, Andrei Makartchouk

144 Handbook of Machine Tool Analysis, Ioan D Marinescu,

Constantin Ispas, and Dan Boboc

145 Implementing Concurrent Engineering in Small Companies,

Susan Carlson Skalak

146 Practical Guide to the Packaging of Electronics: Thermal and

Mechanical Design and Analysis, Ali Jamnia

147 Bearing Design in Machinery: Engineering Tribology and Lubrication,Avraham Harnoy

148 Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing, R E Little

149 Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V Ganapathy

150 The CAD Guidebook: A Basic Manual for Understanding and

Improving Computer-Aided Design, Stephen J Schoonmaker

151 Industrial Noise Control and Acoustics, Randall F Barron

152 Mechanical Properties of Engineered Materials, Wolé Soboyejo

153 Reliability Verification, Testing, and Analysis in Engineering Design, Gary S Wasserman

154 Fundamental Mechanics of Fluids: Third Edition, I G Currie

155 Intermediate Heat Transfer, Kau-Fui Vincent Wong

156 HVAC Water Chillers and Cooling Towers: Fundamentals, Application,and Operation, Herbert W Stanford III

157 Gear Noise and Vibration: Second Edition, Revised and Expanded,

J Derek Smith

158 Handbook of Turbomachinery: Second Edition, Revised and

Expanded, edited by Earl Logan, Jr and Ramendra Roy

159 Piping and Pipeline Engineering: Design, Construction, Maintenance,Integrity, and Repair, George A Antaki

160 Turbomachinery: Design and Theory, Rama S R Gorla

and Aijaz Ahmed Khan

161 Target Costing: Market-Driven Product Design, M Bradford Clifton,Henry M B Bird, Robert E Albano, and Wesley P Townsend

162 Fluidized Bed Combustion, Simeon N Oka

163 Theory of Dimensioning: An Introduction to Parameterizing GeometricModels, Vijay Srinivasan

164 Handbook of Mechanical Alloy Design, edited by George E Totten, Lin Xie, and Kiyoshi Funatani

165 Structural Analysis of Polymeric Composite Materials, Mark E Tuttle

166 Modeling and Simulation for Material Selection and MechanicalDesign, edited by George E Totten, Lin Xie, and Kiyoshi Funatani

167 Handbook of Pneumatic Conveying Engineering, David Mills,

Mark G Jones, and Vijay K Agarwal

168 Clutches and Brakes: Design and Selection, Second Edition,

William C Orthwein

169 Fundamentals of Fluid Film Lubrication: Second Edition,

Bernard J Hamrock, Steven R Schmid, and Bo O Jacobson

170 Handbook of Lead-Free Solder Technology for Microelectronic Assemblies, edited by Karl J Puttlitz and Kathleen A Stalter

171 Vehicle Stability, Dean Karnopp

172 Mechanical Wear Fundamentals and Testing: Second Edition,

Revised and Expanded, Raymond G Bayer

173 Liquid Pipeline Hydraulics, E Shashi Menon

174 Solid Fuels Combustion and Gasification, Marcio L de Souza-Santos

175 Mechanical Tolerance Stackup and Analysis, Bryan R Fischer

176 Engineering Design for Wear, Raymond G Bayer

177 Vibrations of Shells and Plates: Third Edition, Revised and Expanded,Werner Soedel

178 Refractories Handbook, edited by Charles A Schacht

179 Practical Engineering Failure Analysis, Hani M Tawancy,

Anwar Ul-Hamid, and Nureddin M Abbas

180 Mechanical Alloying and Milling, C Suryanarayana

181 Mechanical Vibration: Analysis, Uncertainties, and Control,

Second Edition, Revised and Expanded, Haym Benaroya

182 Design of Automatic Machinery, Stephen J Derby

183 Practical Fracture Mechanics in Design: Second Edition,

Revised and Expanded, Arun Shukla

184 Practical Guide to Designed Experiments, Paul D Funkenbusch

185 Gigacycle Fatigue in Mechanical Practive, Claude Bathias

and Paul C Paris

186 Selection of Engineering Materials and Adhesives, Lawrence W Fisher

187 Boundary Methods: Elements, Contours, and Nodes, Subrata

Mukherjee and Yu Xie Mukherjee

188 Rotordynamics, Agnieszka (Agnes) Muszn´yska

189 Pump Characteristics and Applications: Second Edition,

192 Micro Electro Mechanical System Design, James J Allen

193 Probability Models in Engineering and Science, Haym Benaroya and Seon Han

194 Damage Mechanics, George Z Voyiadjis and Peter I Kattan

195 Standard Handbook of Chains: Chains for Power Transmission and Material Handling, Second Edition, American Chain Associationand John L Wright, Technical Consultant

196 Standards for Engineering Design and Manufacturing,

Wasim Ahmed Khan and Abdul Raouf S.I

197 Maintenance, Replacement, and Reliability: Theory and Applications, Andrew K S Jardine and Albert H C Tsang

198 Finite Element Method: Applications in Solids, Structures, and HeatTransfer, Michael R Gosz

199 Microengineering, MEMS, and Interfacing: A Practical Guide,

Danny Banks

200 Fundamentals of Natural Gas Processing, Arthur J Kidnay

and William Parrish

201 Optimal Control of Induction Heating Processes, Edgar Rapoport

and Yulia Pleshivtseva

202 Practical Plant Failure Analysis: A Guide to Understanding Machinery

Deterioration and Improving Equipment Reliability,

Neville W Sachs, P.E

203 Shaft Alignment Handbook, Third Edition, John Piotrowski

204 Advanced Vibration Analysis , S Graham Kelly

205 Principles of Composite Materials Mechanics, Second Edition,

Ronald F Gibson

206 Applied Combustion, Second Edition, Eugene L Keating

207 Introduction to the Design and Behavior of Bolted Joints,

Fourth Edition: Non-Gasketed Joints, John H Bickford

208 Analytical and Approximate Methods in Transport Phenomena,

Marcio L de Souza-Santos

209 Design and Optimization of Thermal Systems, Second Edition,

Yogesh Jaluria

210 Friction Science and Technology: From Concepts to Applications,

Second Edition, Peter J Blau

211 Practical Guide to the Packaging of Electronics, Second Edition:

Thermal and Mechanical Design and Analysis, Ali Jamnia

212 Practical Stress Analysis in Engineering Design, Third Edition,

Ronald L Huston and Harold Josephs

213 Principles of Biomechanics, Ronald L Huston

214 Mechanical Vibration Analysis, Uncertainties, and Control,

Third Edition, Haym Benaroya and Mark L Nagurka

215 Solid Fuels Combustion and Gasification: Modeling, Simulation,

and Equipment Operations, Second Edition, Third Edition,

Marcio L de Souza-Santos

216 Asset Management Excellence, edited by John D Campbell,

Andrew K S Jardine, Joel McGlynn

217 Mechanical Tolerance Stackup and Analysis, Second Edition,

Bryan R Fischer

218 Principles of Composite Materials Mechanics, Third Edition,

Ronald F Gibson

This page intentionally left blank

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

PRINCIPLES OF COMPOSITE

MATERIAL

MECHANICS

Ronald F Gibson

THIRD EDITION

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2012 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S Government works

Version Date: 20110810

International Standard Book Number-13: 978-1-4398-5006-0 (eBook - PDF)

This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint.

Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.

transmit-For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC,

a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used

only for identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

To my wonderful family, Maryanne, Tracy, Tola, Sophie, and

Aidan, and the memory of my parents,

Jim and Lora Gibson

This page intentionally left blank

Contents

Preface xxi

Preface.to.the.Second.Edition xxiii

Preface.to.the.First.Edition xxv

Author xxix

1 Introduction 1

1.1 Basic.Concepts 1

1.2 Constituent.Materials.for.Composites 6

1.2.1 Reinforcement.Materials, Including.Nanoreinforcements 10

1.2.2 Matrix.and.Filler.Materials 14

1.3 Structural.Applications.of.Composites 15

1.4 Multifunctional.Applications.of.Composites 25

1.5 Fabrication.Processes 28

1.6 Elements.of.Mechanical.Behavior.of.Composites 38

1.7 Review.of.Basic.Mechanics.of.Materials.Equations 40

Problems 47

References 51

2 Lamina Stress–Strain Relationships 53

2.1 Introduction 53

2.2 Effective.Moduli.in.Stress–Strain.Relationships 54

2.3 Symmetry.in.Stress–Strain.Relationships 59

2.4 Orthotropic.and.Isotropic.Engineering.Constants 64

2.5 Specially.Orthotropic.Lamina 67

2.6 Generally.Orthotropic.Lamina 71

Problems 84

References 87

3 Effective Moduli of a Continuous Fiber-Reinforced Lamina 89

3.1 Introduction 89

3.2 Elementary.Mechanics.of.Materials.Models 97

3.2.1 Longitudinal.Modulus 99

3.2.2 Transverse.Modulus 103

3.2.3 Shear.Modulus.and.Poisson’s.Ratio 105

3.3 Improved.Mechanics.of.Materials.Models 108

3.4 Elasticity.Models 113

3.4.1 Finite.Difference.Models 114

3.4.2 Finite.Element.Models 116

3.4.3 Closed-Form.and.Variational.Models 121

xvi Contents

3.5 Semiempirical.Models 123

Problems 126

References 131

4 Strength of a Continuous Fiber-Reinforced Lamina 135

4.1 Introduction 135

4.2 Multiaxial.Strength.Criteria 138

4.2.1 Maximum.Stress.Criterion 140

4.2.2 Maximum.Strain.Criterion 145

4.2.3 Quadratic.Interaction.Criteria 147

4.3 Micromechanics.Models.for.Lamina.Strength 155

4.3.1 Longitudinal.Strength 156

4.3.2 Transverse.Strength 164

4.3.3 In-Plane.Shear.Strength 169

4.3.4 Multiaxial.Strength 169

Problems 173

References 175

5 Analysis of Lamina Hygrothermal Behavior 181

5.1 Introduction 181

5.2 Hygrothermal.Degradation.of.Properties 183

5.3 Lamina.Stress–Strain.Relationships.Including Hygrothermal Effects 195

5.4 Micromechanics.Models.for.Hygrothermal.Properties 202

Problems 211

References 215

6 Analysis of a Discontinuously Reinforced Lamina 219

6.1 Introduction 219

6.2 Aligned.Discontinuous.Fibers 220

6.2.1 Stress.and.Strength.Analysis 221

6.2.2 Modulus.Analysis 229

6.3 Off.Axis–Aligned.Discontinuous.Fibers 239

6.3.1 Stress.and.Strength.Analysis 239

6.3.2 Modulus.Analysis 242

6.4 Randomly.Oriented.Discontinuous.Fibers 245

6.4.1 Stress.and.Strength.Analysis 245

6.4.2 Modulus.Analysis 247

6.5 Nanofibers.and.Nanotubes 258

6.5.1 Stress.and.Strength.Analysis 259

6.5.2 Modulus.Analysis 260

6.6 Particulates 265

6.6.1 Stress.and.Strength.Analysis 266

6.6.2 Modulus.Analysis 267

6.7 Hybrid.Multiscale.Reinforcements 271

Contents xvii

Problems 275

References 278

7 Analysis of Laminates 283

7.1 Introduction 283

7.2 Theory.of.Laminated.Beams 284

7.2.1 Flexural.Stresses.and.Deflections 284

7.2.2 Shear.Stresses.and.Deflections 290

7.3 Theory.of.Laminated.Plates.with.Coupling 296

7.4 Stiffness.Characteristics.of.Selected Laminate.Configurations 304

7.4.1 Specially.Orthotropic.Laminates 305

7.4.2 Generally.Orthotropic.Laminates 305

7.4.3 Symmetric.Laminates 306

7.4.4 Antisymmetric.Laminates 308

7.4.5 Quasi-Isotropic.Laminates 310

7.5 Derivation.and.Use.of.Laminate.Compliances 314

7.5.1 Inversion.of.Laminate.Force–Deformation Equations 315

7.5.2 Determination.of.Lamina.Stresses.and.Strains 316

7.5.3 Determination.of.Laminate.Engineering.Constants 320

7.5.4 Comparison.of.Measured.and Predicted.Compliances 324

7.6 Hygrothermal.Effects.in.Laminates 327

7.6.1 Hygrothermal.Degradation.of.Laminates 327

7.6.2 Hygrothermal.Stresses.in.Laminates 328

7.6.3 Laminate.Hygrothermal.Expansion.Coefficients 332

7.7 Interlaminar.Stresses 333

7.8 Laminate.Strength.Analysis 339

7.8.1 First.Ply.Failure.and.Subsequent.Ply.Failures Due to In-Plane Stresses 340

7.8.2 Delamination.Due.to.Interlaminar.Stresses 352

7.9 Deflection.and.Buckling.of.Laminates 357

7.9.1 Analysis.of.Small.Transverse.Deflections 358

7.9.2 Buckling.Analysis 364

7.10 Selection.of.Laminate.Designs 367

7.11 Application.of.Laminate.Analysis.to.Composite Structures 375

7.11.1 Composite.Sandwich.Structures 376

7.11.2 Composite.Grid.Structures 382

Problems 387

References 395

8 Analysis of Viscoelastic and Dynamic Behavior 399

8.1 Introduction 399

xviii Contents

8.2 Linear.Viscoelastic.Behavior.of.Composites 402

8.2.1 Boltzmann.Superposition.Integrals.for.Creep and Relaxation 405

8.2.2 Differential.Equations.and.Spring– Dashpot.Models 411

8.2.3 Quasielastic.Analysis 421

8.2.4 Sinusoidal.Oscillations.and.Complex Modulus.Notation 424

8.2.5 Elastic–Viscoelastic.Correspondence.Principle 430

8.2.6 Temperature.and.Aging.Effects 436

8.3 Dynamic.Behavior.of.Composites 443

8.3.1 Longitudinal.Wave.Propagation.and.Vibrations.in Specially Orthotropic.Composite.Bars 444

8.3.2 Flexural.Vibration.of.Composite.Beams 450

8.3.3 Transverse.Vibration.of.Laminated.Plates 454

8.3.4 Analysis.of.Damping.in.Composites 464

8.4 Nanoenhancement.of.Viscoelastic.and.Dynamic Properties 476

Problems 479

References 487

9 Analysis of Fracture 493

9.1 Introduction 493

9.2 Fracture.Mechanics.Analyses.of.Through-Thickness Cracks 493

9.2.1 Stress.Intensity.Factor.Approach 496

9.2.2 Strain.Energy.Release.Rate.Approach 502

9.2.3 Virtual.Crack.Closure.Technique 506

9.3 Stress.Fracture.Criteria.for.Through-Thickness.Notches 508

9.4 Interlaminar.Fracture 515

9.5 Nanoenhancement.of.Fracture.Toughness 527

Problems 530

References 534

10 Mechanical Testing of Composites and Their Constituents 541

10.1 Introduction 541

10.2 Measurement.of.Constituent.Material.Properties 542

10.2.1 Fiber.Tests 542

10.2.2 Neat.Resin.Matrix.Tests 545

10.2.3 Constituent.Volume.Fraction.Measurement 549

10.3 Measurement.of.Basic.Composite.Properties 549

10.3.1 Tensile.Tests 550

10.3.2 Compressive.Tests 555

10.3.3 Shear.Tests 561

10.3.4 Flexure.Tests 570

Contents xix

10.3.5 Interlaminar.Fracture.Tests 571

10.3.6 Fiber/Matrix.Interface.Tests 574

10.3.7 Open.Hole.and.Filled.Hole.Tests 576

10.3.8 Bearing.Tests 577

10.3.9 Pull-Through.Tests 579

10.4 Measurement.of.Viscoelastic.and.Dynamic.Properties 581

10.4.1 Creep.Tests 581

10.4.2 Vibration.Tests 585

10.5 Measurement.of.Hygrothermal.Properties 593

10.5.1 Glass.Transition.Temperature.Tests 593

10.5.2 Thermal.Expansion.Tests 594

10.5.3 Moisture.Absorption.Tests 595

Problems 596

References 600

Appendix A: Matrix Concepts and Operations 609

Appendix B: Stress Equilibrium Equations 619

Appendix C: Strain–Displacement Equations 623

Index 627

This page intentionally left blank

Preface

The.goals.for.the.third.edition.are.to.continue.to.present.a.unique.blended.coverage.of.classical.and.state-of-the-art.mechanics.of.composites.technolo-gies,.while.striving.to.incorporate.the.most.worked-out.example.problems.and.homework.problems.of.any.available.textbook.on.mechanics.of.compos-ite.materials A.high.priority.has.been.placed.on.writing.a.textbook.that.is.easily.understood.by.students Pedagogical.improvements.include.a.total.of.81.worked-out.example.problems.(25.of.which.are.new.for.the.third.edition);.177.homework.problems.(25.of.which.are.new);.388.figures.(46.of.which.are.new);.a.new.appendix.on.matrix.concepts.and.operations;.new.coverage.of.particle composites, nanocomposites, nanoenhancement of conventional.fiber.composites,.hybrid.multiscale.composites;.and.additional.coverage.of.finite.element.modeling.and.test.methods Although.the.book.is.written.first.as.a.textbook.covering.classical.methods,.frequent.references.to.current.state-of-the-art composites technology and research findings are included Extensive.references.are.provided,.including.many.current.journal.articles.along.with.classic.and.historical.publications

sors.for.their.encouragement.and.support.over.the.years Most.recently,.my.professional affiliations with Wayne State University (WSU) and the.University of Nevada, Reno (UNR) have provided the necessary positive.environments.for.teaching.and.research.that.made.the.development.of.this.new.edition.possible I.have.been.particularly.motivated.by.the.interactions.with.students.in.my.composites.classes.at.WSU.and.UNR As.with.the.earlier.editions.of.this.book,.my.wife.and.best.friend,.Maryanne,.continues.to.be.my.most.enthusiastic.supporter,.and.there.is.no.way.that.I.could.have.completed.this.project.without.her

I.continue.to.be.indebted.to.my.colleagues,.graduate.students,.and.spon-Ronald F Gibson

This page intentionally left blank

Preface to the Second Edition

There.have.been.many.developments.in.mechanics.of.composite.materials.since.the.first.edition.of.this.book.was.published Accordingly,.the.second.edition.has.new.sections.on.recent.applications.of.composite.mechanics.to.nanocomposites,.composite.grid.structures,.and.composite.sandwich.struc-tures In.order.to.strengthen.the.emphasis.on.the.basic.principles.of.mechan-ics,.I.have.added.a.review.of.the.basic.mechanics.of.materials.equations.in.the.Introduction,.and.appendices.covering.the.derivations.of.stress.equilib-rium equations and strain–displacement relations from elasticity theory Coverage.of.micromechanics.in.Chapter.3.has.been.revised.to.include.more.detailed discussions of elasticity and finite element models Chapter 4 on.strength.analysis.has.been.updated.to.include.results.from.the.World.Wide.Failure.Exercise Chapter.8.has.been.improved.by.adding.a.phenomenologi-cal approach to understanding linear viscoelastic behavior of composites Chapter.9.on.fracture.has.been.updated.to.include.coverage.of.the.finite.ele-ment.implementation.of.the.virtual.crack.closure.technique Chapter.10.on.testing.of.composites.and.their.constituents.has.been.extensively.updated.to.include coverage of both new and revised ASTM standard test methods Finally,.more.example.problems.and.homework.problems.have.been.added.to.most.chapters,.and.new.references.have.been.cited.throughout

dents,.and.sponsors I.am.grateful.to.Wayne.State.University.for.providing.a.nurturing.environment.for.my.teaching.and.research,.and.for.granting.me.sabbatical.leave,.which.were.essential.to.the.completion.of.this.project My.graduate.students.have.been.particularly.helpful.in.identifying.the.inevita-ble.errors.in.the.first.edition,.and.their.thesis.research.findings.have.enabled.me.to.add.important.new.dimensions.in.the.second.edition And.as.with.the.first edition, my wife and best friend, Maryanne, has continued to be my.strongest.supporter,.and.I.will.be.forever.grateful.for.her.love,.encourage-ment,.patience,.and.understanding

As.with.the.first.edition,.I.am.indebted.to.many.colleagues,.graduate.stu-Ronald F Gibson

This page intentionally left blank

Preface to the First Edition

Composite.materials.is.truly.an.interdisciplinary.subject,.and.the.number.of.students.taking.courses.in.this.area.is.steadily.increasing Books.on.the.sub-ject.tend.to.emphasize.either.the.mechanics.or.the.materials.science.aspects

of composites Principles of Composite Material Mechanics is mechanics

ori-ented Composite materials technology is new enough for many working.engineers.who.have.had.no.training.in.this.area,.and.so.a.textbook.in.com-posite.material.mechanics.should.be.useful.not.only.for.the.education.of.new.engineers,.but.also.for.the.continuing.education.of.practicing.engineers.and.for.reference The.high.level.of.interest.in.composite.materials,.the.interdisci-plinary.nature.of.the.subject,.the.need.to.reeducate.practicing.engineers,.and.the.need.for.a.new.composite.mechanics.textbook.at.the.introductory.level.all.led.to.my.decision.to.write.this.book

level.course.in.mechanical.engineering,.which.I.have.taught.for.the.last.15.years Chapters.8.through.10,.along.with.selected.papers.from.technical.jour-nals.and.student.research.projects/presentations,.form.the.basis.of.a.second.one-semester.course,.which.is.taken.only.by.graduate.students,.and.which.I.have.taught.for.the.last.4.years The.book.could.also.be.the.basis.for.a.two-quarter.sequence.by.omitting.some.topics Prerequisities.for.the.course.are.knowledge.of.mechanics.of.materials,.introduction.to.materials.engineering,.and.ordinary.differential.equations,.and.previous.exposure.to.linear.algebra.is.highly.desirable For.some.of.the.graduate-level.material,.earlier.courses.in.advanced mechanics of materials, elasticity, and partial differential equa-tions.are.recommended,.but.not.required

Chapters.1.through.7.form.the.basis.of.a.one-semester.senior/graduate-Some.of.the.basic.elements.of.composite.mechanics.covered.in.this.book.have.not.changed.since.the.first.books.on.the.subject.were.published.in.the.1960s.and.1970s,.and,.where.possible,.I.have.tried.to.use.the.accepted.termi-nology.and.nomenclature For.example,.the.coverage.of.stress–strain.rela-tionships and transformation of properties for anisotropic materials in.Chapter.2.and.the.classical.lamination.theory.in.Chapter.7.is.consistent.with

that of previous textbooks such as the Primer on Composite Materials by Ashton, Halpin, and Petit, and Mechanics of Composite Materials by Jones

However,.rather.than.beginning.the.study.of.laminates.by.jumping.directly.into.classical.lamination.theory,.I.have.concluded.that.a.better.pedagogical.approach.is.to.introduce.first.basic.laminate.concepts.by.using.the.simpler.theory.of.laminated.beams.in.pure.flexure Also,.I.believe.that.the.concept.of

an effective modulus of an equivalent homogeneous material, which had

previously been covered only in advanced books such as Mechanics of Composite Materials.by.Christensen,.is.essential.for.the.proper.development.

of heterogeneous composite micromechanics Thus, effective modulus

xxvi Preface to the First Edition

.concepts.are.emphasized.from.their.introduction.in.Chapter.2.to.their.use.in.the.analysis.of.viscoelastic.and.dynamic.behavior.in.Chapter.8

Although.many.basic.concepts.have.been.presented.in.earlier.textbooks,.numerous new developments in composite mechanics over the last two.decades have made it increasingly necessary to supplement these books.with my.own.notes Thus,.I.have.added.coverage.of.such.important.topics.as.hygrothermal.effects.in.Chapter.5,.discontinuous.fiber.composites.in.Chapter.6, viscoelastic behavior and dynamic behavior in Chapter 8, fracture in.Chapter.9,.and.mechanical.testing.in.Chapter.10 The.coverage.of.experimen-tal.mechanics.of.composites.has.been.expanded.to.include.summaries.of.the.most.important.ASTM.standard.test.methods,.many.of.which.did.not.exist.when.the.early.mechanics.of.composites.books.were.published A.variety.of.example problems and homework problems, a number of them related to.practical.composite.structures,.are.also.included

The.contents.of.this.book.represent.the.cumulative.effects.of.more.than.25.years.of.interactions.with.colleagues.and.students,.and.I.would.be.remiss.if.I.did.not.mention.at.least.some.of.them My.fascination.with.composites.began.in.1965.with.my.first.engineering.position.in.what.is.now.part.of.Oak.Ridge.National.Laboratory.in.Tennessee,.where.I.was.involved.in.the.design.and.development.of.high-speed.rotating.equipment At.that.time.I.realized.that.the.advantages.of.using.composites.in.rotating.equipment.are.numer-ous, as is the case in many other applications My experiences working.with Dean Waters and other colleagues in the mechanical development.group.in.Oak.Ridge.have.had.a.strong.influence.on.my.later.career.decision.to.emphasize.composites.research.and.education My.doctoral.research.on.vibration damping characteristics of composites with Robert Plunkett at.the.University.of.Minnesota.further.cemented.my.desire.to.continue.work-ing.in.the.composites.area.and.ultimately.led.to.my.career.in.university.teaching.and.research

After beginning my academic career at Iowa State University in 1975, I.began a long and productive association with C.T Sun, and later had the.pleasure.of.spending.a.one-year.leave.working.with.C.T and.his.colleagues.Robert.Sierakowski.and.Shive.Chaturvedi.at.the.University.of.Florida I.owe.much of my understanding of composite mechanics to interactions with.them The.notes.leading.to.this.book.were.developed.by.teaching.composite.mechanics courses at Iowa State University, the University of Idaho, the.University.of.Florida,.Michigan.State.University,.and.Wayne.State.University,.and.I.am.indebted.to.the.students.who.took.my.classes.and.helped.me.to

“debug” these notes over the years Most recently, my students at Wayne.State University have been particularly effective at finding the inevitable.errors.in.my.notes Interaction.with.my.graduate.students.over.the.years.has.contributed immeasurably to my understanding of composite mechanics,.and.the.work.of.several.of.those.students.has.been.referred.to.in.this.book I.am.particularly.indebted.to.Stalin.Suarez,.Lyle.Deobald,.Raju.Mantena,.and.Jimmy.Hwang,.all.former.graduate.students.at.the.University.of.Idaho

Preface to the First Edition xxvii

Serious work on this book actually began during a sabbatical leave at.Michigan State University in 1987, and I am indebted to Larry Drzal and.his  colleagues for our many stimulating discussions during that year Particularly important was the interaction with Cornelius Horgan, with.whom.I.team-taught.a.course.on.advanced.mechanics.of.composites Most.recently,.my.collaboration.with.John.Sullivan.and.his.colleagues.of.the.Ford.Scientific.Research.Laboratory.has.proved.to.be.very.rewarding,.and.I.am.indebted.to.John.for.his.careful.review.of.the.manuscript.and.helpful.com-ments I am grateful to Carl Johnson, also of the Ford Scientific Research.Laboratory,.for.his.encouragement.and.support.and.for.providing.several.of.the.figures.in.Chapter.1 The.strong.support.of.Wayne.State.University,.which.made.it.possible.to.establish.the.Advanced.Composites.Research.Laboratory.there.in.1989,.is.gratefully.acknowledged The.support.and.encouragement.of

my department chairman, Ken Kline, has been particularly important Generous.support.for.my.composites.research.from.numerous.funding.agen-cies.over.the.years.has.also.helped.to.make.this.book.possible Grants.from.the.Air.Force.Office.of.Scientific.Research,.the.National.Science.Foundation,.the Army Research Office, the Boeing Company, and the Ford Motor.Company.have.been.particularly.important

McGraw-Hill.and.I.would.like.to.thank.the.following.reviewers.for.their.many helpful comments and suggestions: Charles W Bert, University of.Oklahoma; Olivier A Bauchau, Rensselaer Polytechnic Institute; Shive.Chaturvedi, Ohio State University; Vincent Choo, New Mexico State.University;.John.M Kennedy,.Clemson.University;.Vikram.K Kinra,.Texas

A &.M.University;.C.T Sun,.University.of.Florida;.and.Steven.W Yurgartis,.Clarkson.University

porter.as.I.labored.on.this.project,.and.there.is.no.way.that.I.could.have.done.it.without.her.love,.encouragement,.patience,.and.understanding

Finally,.my.wife.and.best.friend,.Maryanne,.has.been.my.strongest.sup-Ronald F Gibson

This page intentionally left blank

Author

mechanical.engineering.at.the.University.of.Nevada,.Reno He.has.a.PhD.in.mechanics.from.the.University.of.Minnesota,.an.MS.in.mechanical.engineer-ing.from.the.University.of.Tennessee,.and.a.BS.in.mechanical.engineering.from.the.University.of.Florida His.industrial/government.experience.includes.a.position.as.development.engineer.for.Union.Carbide.Corporation,.Nuclear.Division and a summer faculty fellowship at the NASA Langley Research.Center He.has.held.full-time.faculty.positions.at.Iowa.State.University,.The.University.of.Idaho,.and.Wayne.State.University,.as.well.as.visiting.faculty.positions.at.Stanford.University,.the.University.of.Florida,.and.Michigan.State.University He.is.an.elected.Fellow.of.the.American.Society.of.Mechanical.Engineers.and.the.American.Society.for.Composites,.and.a.member.of.the.American Society for Engineering Education, the American Institute for.Aeronautics.and.Astronautics,.the.Society.for.Experimental.Mechanics,.and.the.Society.for.Advancement.of.Material.and.Process.Engineering He.served.as.the.president.of.the.American.Society.for Composites.in.2004–2005,.and.as.vice president during 2002–2003 Dr Gibson’s awards include the Hetenyi.Award.for.Best.Research.Paper.of.the.Year.from the.Society.for.Experimental.Mechanics,.the.College.of.Engineering.Outstanding.Faculty.Award.from.the.University.of.Idaho,.the.Distinguished.Faculty.Fellowship.Award,.the.DeVlieg.Professorship, and the Outstanding Graduate Faculty Mentor Award from.Wayne.State.University The.results.of.his.research.have.been.published.in.numerous.scholarly.journal.articles.and presented.at.a.variety.of.national.and.international meetings His current research interests include mechanical.characterization.of.composite.materials.and.structures,.noise.and.vibration.control.with.composites,.design.and.manufacturing.of.composite.structures,.characterization.of.energy-absorbing.materials,.multifunctional.composites,.and.nanocomposites

This page intentionally left blank

“nanocomposites” having nanometer-sized reinforcements such as carbon.nanoparticles, nanofibers, and nanotubes, because of the extraordinary.properties.of.these.materials.

The.relative.importance.of.the.four.basic.materials.in.a.historical.context.has.been.presented.by.Ashby.[1],.as.shown.schematically.in.Figure.1.1.that.clearly.shows.the.steadily.increasing.importance.of.polymers,.composites,.and.ceramics.and.the.decreasing.role.of.metals Composites.are.generally.used.because.they.have.desirable.properties.that.cannot.be.achieved.by.any.of.the.constituent.materials.acting.alone The.most.common.example.is.the.fibrous.composite.consisting.of.reinforcing.fibers.embedded.in.a.binder.or.matrix.material Particle.or.flake.reinforcements.are.also.used,.but.they.are.generally.not.as.effective.as.fibers

Although it is difficult to say with certainty when or where humans.first  learned about fibrous composites, nature provides us with numerous.examples Wood consists mainly of fibrous cellulose in a matrix of lignin,.whereas most mammalian bone is made up of layered and oriented colla-gen  fibrils in a protein–calcium phosphate matrix [2] The book of Exodus

in the.Old.Testament.recorded.what.surely.must.be.one.of.the.first.examples.of.man-made.fibrous.composites,.the.straw-reinforced.clay.bricks.used.by.the.Israelites The.early.natives.of.South.America.and.Central.America.apparently.used.plant.fibers.in.their.pottery These.early.uses.of.fibrous.reinforcement,.however,.were.probably.based.on.the.desire.to.keep.the.clay.from.cracking.during.drying.rather.than.on.structural.reinforcement Much.later,.humans

2 Principles of Composite Material Mechanics

developed.structural.composites.such.as.steel-.reinforced concrete,.polymers.reinforced.with.fibers.such.as.glass.and.carbon,.and.many.other materials.Fibrous.reinforcement.is.very.effective.because.many.materials.are.much.stronger.and.stiffer.in.fiber.form.than.they.are.in.bulk.form It.is.believed.that.this.phenomenon.was.first.demonstrated.scientifically.in.1920.by.Griffith.[3], who measured the tensile strengths of glass rods and glass fibers of different.diameters Griffith.found.that.as.the.rods.and.fibers.got.thinner,.they.got.stronger.(see.Figure.1.2.from.Ref [3],.as.shown.in.Ref [4]),.appar-ently.because.the.smaller.the.diameter,.the.smaller.the.likelihood.that.failure-inducing.surface.cracks.would.be.generated.during.fabrication.and.handling By.extrapolating.these.results,.Griffith.found.that.for.very.small.diameters,.the fiber strength approached the theoretical cohesive strength between.adjacent layers of atoms, whereas for large diameters, the fiber strength.dropped.to.nearly.the.strength.of.bulk.glass

Results.similar.to.those.published.by.Griffith.have.been.reported.for.a.wide.variety.of.other.materials The.reasons.for.the.differences.between.fiber.and.bulk.behavior,.however,.are.not.necessarily.the.same.for.the.other.materials For example, polymeric fibers are stronger and stiffer than bulk .polymers

Iron Cast iron

Rubber Glues

PE PMMA

PC PS PP CFRP

Ceramic composites

Tough engineering ceramics (Al2O3Si3N4 PCZ etc.) CFRP

AcrylicsEpoxies

Metal-matrix Refractories

Portland cement Fusedsilica Cemets Pyro,

Alloys

Light alloys Super alloys Titanium zirconium etc.

New super alloys

Conducting polymers High-temperature polymers

Microalloyed steels Dual phase steels

Glassy metals Al–lithium alloys

composites

Development stow: Mostly quality control and processing

Composites

Composites

Ceramics Polymers Metals

High-modulus polymers Polyesters Kevlar®FRP

FIGURE 1.1

The.relative.importance.of.metals,.polymers,.composites,.and.ceramics.as.a.function.of.time The.diagram.is.schematic.and.describes.neither.tonnage.nor.value The.timescale.is.nonlinear (Ashby, M F 1987 Technology of the 1990s: Advanced materials and predictive design

Philosophical Transactions of the Royal Society of London sion.of.The.Royal.Society.)

,.A322,.393–407 Reproduced.by.permis-Introduction 3

because.of.the.highly.aligned.and.extended.polymer.chains.in.the.fibers.and.the.randomly.oriented.polymer.chains.in.the.bulk.polymer A.similar.effect.occurs.in.crystalline.materials.such.as.graphite In.addition,.a.single.crystal.tends.to.have.a.lower.dislocation.density.than.a.polycrystalline.solid;.so.sin-gle-crystal.“whisker”.materials.are.much.stronger.than.the.same.material.in.polycrystalline.bulk.form Whiskers.typically.have.dimensions.in.the.microm-eter.range,.and.for.many.years,.it.was.thought.that.whiskers.were.the.stron-gest.and.stiffest.reinforcement.materials.available However,.it is.now.believed.that.carbon.nanotubes,.which.have.dimensions.in.the.nanometer.range,.are.the.strongest.and.stiffest.reinforcement.materials.in.existence.[5,6]

There.can.be.no.doubt.that.fibers.allow.us.to.obtain.the.maximum.tensile.strength.and.stiffness.of.a.material,.but.there.are.obvious.disadvantages.of.using.a.material.in.fiber.form Fibers.alone.cannot.support.longitudinal.com-pressive.loads.and.their.transverse.mechanical.properties.are.often.not.as.good.as.the.corresponding.longitudinal.properties Thus,.fibers.are.gener-ally.useless.as.structural.materials.unless.they.are.held.together.in.a.struc-tural unit with a binder or matrix material and unless some transverse.reinforcement is provided Fortunately, the geometrical configuration of.fibers.also.turns.out.to.be.very.efficient.from.the.point.of.view.of.interaction.with.the.binder.or.matrix As.shown.in.Figure.1.3.from.Ref [7],.the.ratio.of.surface.area.to.volume.for.a.cylindrical.particle.is.greatest.when.the.particle.is.in.either.platelet.or.fiber.form For.a.platelet,.the.particle.aspect.ratio,.that

Extrapolates to 1.600.000 psi (11.000 MN/m 2 ) 600,000

500,000 400,000 300,000

Thickness of fiber, in

20 μ

10 μ 1

1000 10002 10003 10004

FIGURE 1.2

Griffith’s measurements of tensile strength as a function of fiber thickness for glass fibers

(Griffith,.A A 1920 The.phenomena.of.rupture.and.flow.in.solids Philosophical Transactions of

the Royal Society,.221A,.163–198 Reproduced.by.permission.of.The.Royal.Society;.Gordon,.J E

1976 The New Science of Strong Materials, 2nd ed Princeton University Press, Princeton, NJ

With.permission.)

4 Principles of Composite Material Mechanics

is,.a.=.l/d.(i.e.,.the.length-to-diameter.ratio).is.very.small,.whereas.for.a.fiber,.

the.aspect.ratio.is.very.large Thus,.the.fiber/matrix.interfacial.area.available.for.stress.transfer.per.unit.volume.of.fiber.increases.with.increasing.fiber.length-to-diameter.ratio It.is.also.important.to.note.that,.for.a.fiber- reinforced.composite,.the.total.fiber/matrix.interfacial.area.will.be.increased.if.the.fiber.diameter is decreased while maintaining constant fiber volume and fiber.length

Example 1.1

Compare the total fiber surface area of a group of N small-diameter fibers with

that of a single large-diameter fiber having the same length and volume.

SOLUTION

Assuming that the fibers are perfectly round, for a single large-diameter fiber of

diameter dL, the circumferential fiber surface area per unit length is

while the fiber volume per unit length is

VL = πdL24

20

15 10 5

Engineering,.1988,.by.permission.of.Oxford.University.Press,.New.York.).

For N small-diameter fibers having the same volume and length as a single

large-diameter fiber,

so that

N V V

d d

= L =  

S L S 2

The ratio of the total surface area of N small-diameter fibers to the surface area

of a single large-diameter fiber having the same volume and length is then

R NA A

Nd d

d d

d d

d d

For example, if a large fiber is replaced by a group of fibers having diameters

1000 times smaller, dL = 1000 d S , and the total fiber surface area for a constant fiber length and volume will increase by a factor of 1000 It can be easily shown that the same result holds for spherical particles So in addition to their superior strength and stiffness, small-diameter fibers such as nanofibers have the additional advantage of significantly larger fiber/matrix interfacial surface area per unit vol- ume than larger-diameter fibers.

ronmental.attack Transverse.reinforcement.is.generally.provided.by.orient-ing.fibers.at.various.angles.according.to.the.stress.field.in.the.component.of.interest Filler.particles.are.also.commonly.used.in.composites.for.a.variety.of.reasons.such.as.weight.reduction,.cost.reduction,.flame.and.smoke.suppres-sion,.and.prevention.of.ultraviolet.degradation.due.to.exposure.to.sunlight.The.need.for.fiber.placement.in.different.directions.according.to.the.par-ticular.application.has.led.to.various.types.of.composites,.as.shown.in.Figure.1.4 In.the.continuous.fiber.composite.laminate.(Figure.1.4a),.individual.con-tinuous fiber/matrix laminae are oriented in the required directions and.bonded.together.to.form.a.laminate Although.the.continuous.fiber.laminate.is.used.extensively,.the.potential.for.delamination,.or.separation.of.the.lami-nae,.is.still.a.major.problem.because.the.interlaminar.strength.is.matrix.domi-nated Woven.fiber.composites.(Figure.1.4b).do.not.have.distinct.laminae.and.are.not.susceptible.to.delamination,.but.strength.and.stiffness.are.sacrificed

The.matrix.also.serves.to.protect.the.fibers.from.external.damage.and.envi-6 Principles of Composite Material Mechanics

because the fibers are not as straight as in the continuous fiber laminate Chopped.fiber.composites.may.have.short.fibers.randomly.dispersed.in.the.matrix,.as.shown.in.Figure.1.4c Chopped.fiber.composites.are.used.exten-sively.in.high-volume.applications.due.to.their.low.manufacturing.cost,.but.their.mechanical.properties.are.considerably.poorer.than.those.of.continuous.fiber.composites Finally,.hybrid.composites.may.consist.of.mixed.chopped.and.continuous.fibers,.as.shown.in.Figure.1.4d,.or.mixed.fiber.types.such.as.glass.and.carbon Another.common.composite.configuration,.the.sandwich.structure.(Figure.1.5),.consists.of.high-strength.composite.facing.sheets.(which.could.be.any.of.the.composites.shown.in.Figure.1.4).bonded.to.a.lightweight.foam.or.honeycomb.core Sandwich.structures.have.extremely.high.flexural.stiffness-to-weight.ratios.and.are.widely.used.in.aerospace.structures The.design.flexibility.offered.by.these.and.other.composite.configurations.is.obvi-ously.quite.attractive.to.designers,.and.the.potential.exists.to.design.not.only.the.structure.but.also.the.structural.material.itself

Introduction 7

such as automotive vehicles because of their low cost, and are sometimes.referred.to.as.“basic”.composites The.so-called.“advanced”.composites.made.from.carbon,.silicon.carbide,.aramid.polymer,.boron,.or.other.higher-modulus.fibers are used mainly in more demanding applications such as aerospace.structures.where.their.higher.cost.can.be.justified.by.improved.performance.The.tremendous.advantages.of.advanced.fibers.over.glass.fibers.and.con-ventional bulk materials are shown in Table 1.1, by comparing selected.properties The.main.advantages.are.higher.modulus,.higher.strength,.and.lower density In many applications, such as aerospace and automotive.structures,.structural.weight.is.very.important Depending.on.whether.the.structural.design.is.strength-critical.or.stiffness-critical,.the.material.used.should.have.a.high.strength-to-weight.ratio.(or.specific.strength).or.a.high.stiffness-to-weight ratio (or specific stiffness) As shown in Figure 1.6,.advanced.fibers.also.have.significant.advantages.over.conventional.materi-als.in.both.these.attributes,.and.this.is.the.principal.reason.that.composites.will be used with increasing frequency in aerospace structures, fiber-.reinforced.automotive.structures,.and.other.structures.where.these.proper-ties.are.important When.the.reinforcing.fibers.are.combined.with.a.matrix.or.binder.material.to.form.a.composite,.there.is.obviously.some.reduction.in.the.tremendously.high.specific.strengths.and.specific.moduli.shown.for.the.reinforcing.fibers.in.Figure.1.6,.but.the.composites.still.have.an.overwhelm-ing.advantage.with.respect.to.these.properties Table.1.1.also.shows.that.the.ultimate.strain,.or.strain.to.failure.(a.measure.of.ductility),.is.one.attribute.where reinforcing fibers are typically not as good as conventional bulk

Film adhesive bonds

facings to core

High strength composite laminate facings

Low density

honeycomb core

FIGURE 1.5

Composite.sandwich.structure.

Tensile Modulus (GPa)

Density (g/cm 3 )

Specific Strength (MPa/ [g/cm 3 ])

Specific Modulus (GPa/[g/cm 3 ])

Ultimate Strain Manufacturer